Maurer--Cartan structure equation

Bilinearity of the bracket [F5] and smoothness of $\theta$ [F3] show in any basis of $\mathfrak{g}$ that the displayed bracket-wedge has smooth components; alternation follows by exchanging $A$ and $B$. Thus it is a well-defined $\mathfrak{g}$-valued two-form. Because the bracket is alternating, [F5] also gives $[\theta \wedge \theta ](A,B)=2[\theta (A),\theta (B){]}_G$.

Let $X,Y\in \mathfrak{g}$ and use their left-invariant extensions from [F6]. For every $\lambda \in {\mathfrak{g}}^{\ast }$, [F2], [F7], and [F3] give $\lambda (d\theta (X^L,Y^L))=X^L(\lambda (\theta (Y^L)))-Y^L(\lambda (\theta (X^L)))-\lambda (\theta ([X^L,Y^L]))=-\lambda ([X,Y{]}_G),$ because the first two functions are the constants $\lambda (Y)$ and $\lambda (X)$ and [F4] identifies the bracket field. Since linear functionals separate points, $d\theta (X^L,Y^L)=-[X,Y{]}_G$.

On the same fields, step 1.1 and [F3] yield $\frac{1}{2}[\theta \wedge \theta ](X^L,Y^L)=[X,Y{]}_G$. Adding this to step 1.2 proves the structure equation on every pair of left-invariant fields.

Fix $p\in G$ and $U,V\in T_{p}G$. Put $X={\theta }_p(U)$ and $Y={\theta }_p(V)$. By [F3], ${\theta }_p(X_p^L)=X={\theta }_p(U)$ and ${\theta }_p(Y_p^L)=Y={\theta }_p(V)$; injectivity of ${\theta }_p$ gives $X_p^L=U$ and $Y_p^L=V$. Step 2.1 therefore makes the structure equation vanish on the arbitrary pair $(U,V)$ at $p$, proving it globally.

A Lie group is nonempty. If $\dim G=0$, both two-forms are uniquely zero; if $\dim G=1$, every alternating two-form is zero and the equation again holds. Lie groups are boundaryless by convention, and no metric, nondegeneracy, or endpoint is involved. The stated ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited through [F3], [F4], [F5], and [F6]; componentwise differentiation and the pointwise spanning argument add no choice. The theorem is one equality, not a biconditional.